#9863448

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,9)(3,6,4,8)(5,10,14,16)(11,13,12,15)(17,18)(19,24)(20,21)(22,31)(23,27)(25,30)(26,28)(29,32)(33,34,39,36)(35,42,37,40)(38,44,46,41)(43,47,45,48);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,23,29)(18,24,27,32)(20,28,30,22)(21,26,25,31)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(2,9)(3,4)(10,16)(11,12)(17,20,23,30)(18,25,27,21)(19,22,29,28)(24,26,32,31)(33,35,39,37)(34,40,36,42)(38,43,46,45)(41,47,44,48);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20)(18,21)(19,28)(22,29)(23,30)(24,26)(25,27)(31,32)(33,36,39,34)(35,40,37,42)(38,44,46,41)(43,47,45,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21)(18,20)(19,26)(22,32)(23,25)(24,28)(27,30)(29,31)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,22,23,28)(18,26,27,31)(19,20,29,30)(21,24,25,32)(33,38,39,46)(34,41,36,44)(35,43,37,45)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770644);